Number Theory

Moments of zeta and L-functions on the critical Line II (3 of 3)

Speaker: 
K. Soundararajan
Date: 
Fri, Jun 3, 2011
Location: 
University of Calgary, Calgary, Canada
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
Abstract: 

I will discuss techniques to get upper and lower bounds for moments of zeta and L-functions. The lower bounds are unconditional and the upper bounds in general rely on the Riemann Hypothesis. In several cases of low moments, one can obtain asymptotics, and I may discuss a couple of such recent cases.

This lecture is part of a series of 3

  1. Lecture 1: distribution-values-zeta-and-l-functions-1-3
  2. Lecture 2: Moments of zeta and L-functions on the Critical Line, I
  3. Lecture 3: Moments of zeta and L-functions on the critical line, II
Class: 
Scientific
Subject: 
Mathematics
Number Theory

Moments of zeta and L-functions on the critical Line I (2 of 3)

Speaker: 
K. Soundararajan
Date: 
Thu, Jun 2, 2011
Location: 
PIMS, University of Calgary
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
Abstract: 

I will discuss techniques to get upper and lower bounds for moments of zeta and L-functions. The lower bounds are unconditional and the upper bounds in general rely on the Riemann Hypothesis. In several cases of low moments, one can obtain asymptotics, and I may discuss a couple of such recent cases.

This lecture is part of a series of 3

  1. Lecture 1: distribution-values-zeta-and-l-functions-1-3
  2. Lecture 2: Moments of zeta and L-functions on the Critical Line, I
  3. Lecture 3: Moments of zeta and L-functions on the critical line, II
Class: 
Scientific
Subject: 
Mathematics
Number Theory

Distribution of Values of zeta and L-functions (1 of 3)

Speaker: 
K. Soundararajan
Date: 
Thu, Jun 2, 2011
Location: 
PIMS, University of Calgary
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
Abstract: 

I will discuss the distribution of values of zeta and L-functions when restricted to the right of the critical line. Here the values are well understood by probabilistic models involving “random Euler products”. This fails on the critical line, and the L-values here have a different flavor here with Selberg’s theorem on log normality being a representative result.

This lecture is part of a series of 3

  1. Lecture 1: distribution-values-zeta-and-l-functions-1-3
  2. Lecture 2: Moments of zeta and L-functions on the Critical Line, I
  3. Lecture 3: Moments of zeta and L-functions on the critical line, II
Class: 
Scientific
Subject: 
Mathematics
Number Theory

Special values of Artin L-series (3 of 3)

Speaker: 
Ram Murty
Date: 
Thu, Jun 2, 2011
Location: 
PIMS, University of Calgary
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
Abstract: 

Dirichlet’s class number formula has a nice conjectural generalization in the form of Stark’s conjectures. These conjectures pertain to the value of Artin L-series at s = 1. However, the special values at other integer points also are interesting and in this context, there is a famous conjecture of Zagier. We will give a brief outline of this and display some recent results.

This lecture is part of a series of 3.

Class: 
Scientific
Subject: 
Mathematics
Number Theory

Artin’s holomorphy conjecture and recent progress (2 of 3)

Speaker: 
Ram Murty
Date: 
Thu, Jun 2, 2011
Location: 
PIMS, University of Calgary
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
Abstract: 

Artin conjectured that each of his non-abelian L-series extends to an entire function if the associated Galois representation is nontrivial and irreducible. We will discuss the status of this conjecture and discuss briefly its relation to the Langlands program.

This lecture is part of a series of 3.

Class: 
Scientific
Subject: 
Mathematics
Number Theory

Introduction to Artin L-series (1 of 3)

Speaker: 
Ram Murty
Date: 
Thu, Jun 2, 2011
Location: 
PIMS, University of Calgary
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
Abstract: 

After defining Artin L-series, we will discuss the Chebotarev density theorem and its applications.

This lecture is part of a series of 3.

Class: 
Scientific
Subject: 
Mathematics
Number Theory

Expanders, Group Theory, Arithmetic Geometry, Cryptography and Much More

Speaker: 
Eyal Goran
Date: 
Tue, Apr 6, 2010
Location: 
University of Calgary, Calgary, Canada
Abstract: 

This is a lecture given on the occasion of the launch of the PIMS CRG in "L-functions and Number Theory".

The theory of expander graphs is undergoing intensive development. It finds more and more applications to diverse areas of mathematics. In this talk, aimed at a general audience, I will introduce the concept of expander graphs and discuss some interesting connections to arithmetic geometry, group theory and cryptography, including some very recent breakthroughs.

Class: 
Scientific
Subject: 
Mathematics
Algebraic Geometry
Group Theory
Number Theory

Equidistribution and Primes

Author: 
Peter Sarnak
Date: 
Sat, Sep 1, 2007
Location: 
University of British Columbia, Vancouver, Canada
Conference: 
PIMS 10th Anniversary Lectures
Abstract: 

We begin by reviewing various classical problems concerning the existence of primes or numbers with few prime factors as well as some of the key developments towards resolving these long standing questions. Then we put the theory in a natural and general geometric context of actions on affine n-space and indicate what can be established there. The methods used to develop a combinational sieve in this context involve automorphic forms, expander graphs and unexpectedly arithmetic combinatorics. Applications to classical problems such as the divisibility of the areas of Pythagorean triangles and of the curvatures of the circles in an integral Apollonian packing, are given.

Notes: 
PDF icon Sarnak.pdf
Class: 
Scientific
Subject: 
Number Theory

On Hilbert's Tenth Problem

Author: 
Yuri Matiyasevich
Date: 
Tue, Feb 1, 2000
Location: 
University of Calgary, Calgary, Canada
Conference: 
PIMS Distinguished Chair Lectures
Abstract: 

Dr. Matiyasevich is a distinguished logician and mathematician based at the Steklov Institute of Mathematics at St. Petersburg. He is known for his outstanding work in logic, number theory and the theory of algorithms.

At the International Congress of Mathematicians in Paris in 1900 David Hilbert presented a famous list of 23 unsolved problems. It was 70 years later before a solution was found for Hilbert's tenth problem. Matiyasevich, at the young age of 22, acheived international fame for his solution.

Notes: 
PDF icon Matiyasevich.pdf
Class: 
Scientific
Subject: 
Number Theory
Algebra

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